Q . CORNELL UNIVERSITY LIBRARY GIFT OF James lilston pan^ ITBOCT THB Cornell University Library HG8853 .W71 Cost of insurance. olin 3 1924 030 201 085 DATE DUE Jll^llli ^'1" - GAYLORD PRINTED IN U.SA .1^^ Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924030201085 (Zr€4A.eXy dr COST OF INSUEAI^OE. A THEATISK UPON THE COST OF LIFE mSURANCE, TOGETHER WITH AN Arithmetical Explanation of the Computa- tion OF Premiums and Valuation OF Policies. TO "WHICH ARE ADDED TABLES OF NET PEEMTUMS, COST OP INSUEAUOE, &o., FOE THE USE OF lifk insurance agents. NATHAN WTLLEY, Actuary. NEW TORK: Stephen Rnulish, Editor Insurance Times, 1.B7 Broadway, 1870. Entered according to Act of Congress in the year IS70, by STEPHEN ENGLISH, in tlio Clork's Oflieo of the District Conrt of the TTnited States for the Southern District of New York. 124-l^i^ PREFACE. The object of this treatise is to explain the science of life insurance, so that its main features can be easily understood bj"- any one having an ordinary knowledge of arithmetic. The author has avoided as far as possible the algebraic terms which have hitherto made the subject almost incomprehensible to agents ; and he has endeavored to explain and illustrate it in the simplest possible manner. The great want among life insurance companies at the present time is agents who thorougbly understand the nature of the business they are advocating, and who can present arguments based upon the scientific nature of the subject, and meet objections without referring to the home offices for information. The importance of life insurance is now universally admitted by all intelligent men; the information which they require is, whether it will pay, and .what are the necessary premiums which the company must charge for insuring their lives. If they are satisfied that it is a good investment, the victory is half won. About one-half of the tables in this treatise are from a larger work on the " Cost of Insurance," published by Stephen English, Esq., Editor of The Insnravce Timea. The remainder were computed by the author. N. "W". New York, May^ 1870. COST OF INSURANCE. All calculations in life insurance are based upon two simple data : the rate of interest on the invested assets of the compa- nies, and the average rate of mortality among the insured. Both of these are variable qaantities ; the rate of interest varies in different States according to the statute laws and the general demand for money; the rate of mortality varies in different countries and in successive generations, owing to the habits of living, progress in medical science, and the general intelligence of the people, all of which exert a great influence on the average duration of human life. And yet, with all this uncertainty in the fundamental data, life insurance companies can do strict mathematical justice to the insured. This is accomplished by assuming certain fixed and invariable standards of interest and mortality; the former taken so as to come within the lowest average rate, and the latter taken as nearly as possible to the actual experience of companies, and then returning to the insured the surplus which i-emains after the future liabilities on the poli- cies and the present expenses of the business are prtn-ided for. KATE UP INTEREST. The standard rate of interest vvliit'h the assets of coinjianies, or the premiums and the interest thereon, will yield, in future, is established by the. laws of the State of Massachusetts to be four and by New York to be four and a-half per cent. A lower rate of interest than the common one is assumed, because the general experience of life insurance companies has been that when this COST OF INSURANCE. precaution luis not been taken the compauios eveiitually lind tliemselves in\'olved in hopeless baukraptcy. In life insurance all calculations and plans should be made to endui-e throughout a long series of )'ears. A rate of interest on the reserves should be assumed low enough to be perfectly safe ibr the company, and which will, in all probability, come within the limits of any perma- nent change in the value of money. A life insurance com- pany, in consideration of fixed and definite payments, promises to assume responsibilities and to pay obligations which may mature at any future time. The contract made between the policy- holder and the company is binding during the lifetime of the former, and is not to be modified in favor of the company by any financial disturbances which may occur. In a business which has such an element of perpetuity as life insurance, the introduction of even a trivial error in the fundamental data, the assumption of a rate of interest too high, or a table of mortality too low, would produce results which, multiplied by every policy and by every year of business, would ruin any company. For this reason life insurance companies should assume a rate of interest low enough to cover any ordinary fluctuation in the current rates, and return the surplus to the policy-holders in equitable proportions, either in cash or paid-up insurance.- RATE OF MORTALTI^-. The rate of mortality is a basis for life insurance calculations much more difficult to determine than the rate of interest. To persons of ordinary observation, there is nothing so uncertain oi- unforeseen as death, occurring as it does at all ages and I'esnlt- ing from an infinite variety of causes. ' The tables of mortality in common use have been computed after a long and careful observation, and it has been found that in a large number of people of any given age there is a certahr annual ratio of deaths to the numbei' of the living, and that this ratio increases or decreases from year to \-ear with a renui.rkable regularity. COST OF INSURANCE. Among a class of a lew hundred persons we may predict, with a reasonable certainty, that some will die within a year. In a class of one hundred thousand persons living at-a certain age we can predict, with a great deal of accuracy, how many of them will die each successive year until lione of them survive. Numerous tables of mortality have been made, both in this country and in Europe, based upon actual obsei-vatiou and the records of insurance companies, and the rate of mortality in the corresponding ages agrees with great uniformity. The "Actu- aries', or Combined Experience Table of Mortality," has many claims to popularity, being compiled from the mortuary expe- rience of seventeen English life insurance companies ; and it has this advantage, which is possessed by no other table which preceded it, all the data were derived from selected lives ; and for this reason it more accurately represents the experience ol our life insurance companies. As this table has been adopted by the State of Massachusetts, and b\- many of the companies in the United States, as a basis upon which premium raters ;ire calculated, valuations of policies made, and dividends declared, it will be used in all the tables and explanations employed in this treatise. The " American Experience" table of mortality, prepared by Mr. Sheppard Homans, Actuary of the Mutual Life, of New York, from the mortuary records of that company, and graduated by comparing it with the "Combined Experience" table, has been adopted, with four and a-half per cent, interest, as the standard of valuation in the insurajice departments of New York, Missouri, and Michigan, and many companies have made it the basis of their premium rates. EXPLANATION OP THE TERM "COST OP INSURANCE." The observations made in different countries during the past two or three centuries on the mortality of persons at different ages have established the fact that there is a certain law or scale COST OF INSURANCE. Go which the average duration of human life conforms. For all practical purposes in life insurance, it may be regarded as certain that all lives will terminate within a century after birth, but the time when any individual life will end is wholly a matter uf uncertainty. It is this contingency that death may happen which makes life insurance desirable, and as this contingency increases after the days of childhood are passed, the more hazardous and expensive the risk becomes. In fire insurance the ratio of buildings in any class of risks which will burn during a year can be calculated with considerable accurac}', and the whole number insured are taxed to pay the claims which arise from those which are consumed. The annual rates of premium in fire insurance are graded according to the hazard of the risk, and in life insurance the same general principles are adopted. Thus, at the age ten in the " Actuaries' Table of Mortality,'' we have a class of 100,000 lives, and at the end of the first year 676 are assumed to have died, leaving 99,324 survivors. If these 100,000 persons had insured each other for one year in the sum of $1,000 each, the uniform premium required (without regarding expenses and interest) would have been °^^„y or $6.76. At the age twenty there are 93,268 living, and 680 deaths at the end of the year. The premium in this example would be -^"gg.flg— = $7.29. At thirty years of age, we find the premium to be --^^^ = $HA2. At the age seventy the premium has increased to 2=5^^^°=$64.93. This illustration of the net cost of insurance, as applied to temporary policies for only one year, is probably the simplest and the most elementary one which can be given. In practice, however, it is modified by the assumed rate of interest on the invested assets of the company and the kind of policy issued. COST OF INSURANCE. TABLE I. Rate of Mortality, ACCOBDING TO THE "ActuariesV or "Combined Experience" Table. u> DA M 1 6 t 6 M t 1 i a ■ > o es ■4 ^ ^ column ; and M, computed like the K column, is the sums of the discounted numbers of the dying at the various ages. In computing the C column we use a power of -j-.Lj. greater by one than the corresponding age, since the losses by death are regarded as taking place at the end of the year. The numbers in these columns have the same relative value to each othei', and the same results are produced as if we took the first power of y.^5 for the age ten, or any required age, and so on. In Table III., D at the age ten is found by multiplying 100,000, the number of living, by the tenth power of j.i^j ; art the age eleven the number of survivors, 99,324, is multiplied by the eleventh power of j.Vti «i.nd so on. X at the age of ninety-nine is the same as D. At ninety-eight it is found by adding D ninety- iiine and D ninety-eight together, as .085663 + .020592 =.106255. X ninety-seven is obtained by adding N ninety-eight and D ninety-seven together, as .106255 + .28954=. 395795. The C and M columns are found in the same way as D and N noting the difference alluded to in the definition of C above. 14 COST OF mSURANCE. TABLK in. Commutation Columns. Combined Experience Table- Four per cent. ^A^ Age. »., ^. ™a: Age, »» N. 10 11 12 13 14 67,556.41 64,519.00 61,616.50 58 843.(17 56,192.36 1,381,771.74 1,314,215.33 1,249,696.33 1,1881179.83 1,129 236.76 14 411.3725 13,972.2557 13,551.2772 13,147.6913 12,760.2065 65 56 67 58 59 7,340.55 6,905,30 6,486.17 6,082.78 5,604.50 87,924.26 80,583.71 73,678.41 67,19i.24 61,109,46 15 16 17 18 19 63,658.54 51,236.51 48,920.90 46,707.10 44,500.50 1,073,044.40 1,019,385.86 968,149.35 919,228.45 872,521.35 12,387.6229 12,029,3704 11,684.3835 11,352.1716 11,031,7878 60 61 62 63 64 6,320.82 4,960.97 4.614 60 4,281.28 3,960.85 55,414.96 50 094.14 45,133.17 40,518.57 36,237.29 20 21 22 23 24 42,566.31 40,f30,73 38,779.81 37,009.94 35,317.31 827,931.06 785,364.74 744,734.01 7ii5,954.20 668,944.26 10.722.8137 10,424.4069 10,136.2113 9.857.8830 9,588.6992 65 66 67 68 69 3,653 02 3.357.68 3,074 82 2,804.37 2,.546,50 32,276.44 28,623.42 25,265.74 22,190.92 19,386.55 25 26 27 28 29 33,698.63 32,150.76 30,670.39 29,254.65 27,900.63 633,626.95 599,928.32 567,777.56 537,107.17 507.852.52 9,328.3681 9,076.6069 8,832.7948 8,596.6926 8,367,7476 70 71 72 73 74 2,301.43 2,069 22 1,850.05 1,644.05 1,451.37 16,640.05 14,538.62 12,469.40 10.619.35 8,975.298 30 31 32 33 34 26,605.44 25,366.64 24,181.76 23,048.32 21,964.18 479,951.99 453,346.55 427,979.91 403,798.15 380,749.83 8,145,7580 7,930.2313 7,720.9987 7,517 6207 7,319.9565 75 76 77 78 79 1 272.087 1,106.275 953.972 815.0323 669.2945 7,523.928 6,251.811 5,145.566 4,I9I..594 3,376.5616 35 36 37 38 39 20,927 31 19,935.52 18,986.96 18,079.84 17,212.26 358,785.65 337,856.34 317,922.82 298,935,86 280,856,02 7.127.8675 6,940.9785 6 759.1591 6,582.3100 6,410.0966 80 81 82 83 84 576.5783 476.5807 388.83S9 312.8680 247.9142 2,687.2671 2,110.6888 1,634.1281 1,245.2892 932.4212 40 41 42 43 44 16,382.57 15,589.24 14,830.58 14,104.82 13,409.74 263,643.77 247,261.20 231,671.96 216,841. .38 202,736.56 6,242.4238 6,079.1972 5,920.1302 5,764.7740 5,612.1882 85 86 87 88 89 193.1637 147.6411 110.3786 80.4243 56.81712 684.5070 491.3433 343.7022 233.3236 152.8993 45 46 47 48 49 12,743.16 12,103.41 11,488.47 10,897.30 10,328.76 189,326.62 176,683.66 164,480.25 152,991.78 142,094.48 5,461.3623 5,311.7282 5,162.3094 6,013.0062 4,863.5918 90 91 92 93 94 38.65848 25.13852 15.44572 8.83283 4.60963 96.08221 57.4i373 32.28521 16.83949 8.000655 50 51 52 53 54 9,781.92 9,255.78 8 749.40 8,261.90 7,792.46 131,765.72 121,983.80 112,728.02 103,978.62 96,716,72 4,714.0142 4,564.1010 4,413.7091 4,262 7218 4,111.0465 95 06 97 98 99 2.14399 0.85704 0.28954 0.085663 0.020^92 3,396825 1.262835 0.395795 0.106255 0.020592 OT, 3,958.8437 3,805.9337 3,652.3821 3,498.4644 3,344.1394 3,189.4760 3,034.2715 2,876.7084 2,722.8749 2,567.1031 2,411.6189 2,256.7808 2,1113.0580 1,950.8717 1 800.8653 1,653.7385 1,510.0474 1,370.4580 1,235.6094 1,106.1668 982.7057 86,^.8203 756.0657 653.8171 559.4266 473.2219 395.3803 325.9878 264.9724 212.0519 166.8365 128.7433 97.1594 71.4503 50.9364 34.9630 22.9294 14.2039 8.1851 4.3018 2.0133 0.6088 0.2743 0.0816 0.0198 COST OF INSURANCE. 15 COMPUTATION OF PEEMTTTMS. The method of computing premiums will be more easilj' understood if we commence with the ta.ble of mortality (Table I.), and take an example in which the element of interest is omitted. Thus, at the age of ninety-nine, according to the "Combined Experience" table, there is one survivor in the original number of 100,000, and his life is limited to one year. Tf he should insure for $1,000 payable at death, it is evident that the pre- mium would be !i^l,000, since only one premium would be paid. Suppose there are four per.sons at the age of ninetj'-eight who insure for $1,000 each, the policies amounting to $4,000. The first year they pay four premiums in advance, and there are three deaths and three claims 'of ,'t>l,000 each to pay. At the beginning of the second year there will be one premium paid, and during the year there will be one death and one claim of .$1,000. Hence, there would be Hvo premiums paid to secure $-1,000 insurance, and the average premiums would be the sum .)f the dying at the end of each year divided by the sum of the number of the living at the beginning of each j'ear, and the quotient multiplied by $1,000, or the whole amount of claims divided by the whole number of premiums, or 1,000 x(3 + l)-4- ( 4 -f- ] ) = f of 1 , = $ 8 0, the annu al premium required. The annual premium at the age ninety-seven is found in the same manner. The number of persons living is thirteen. The first year there are thirteen premiums paid, the second four, the third one — total eighteen. Tlic number of claims during this time would be thirteen; annual premium, i-3ji.L.ooi— 722.22. The following rule will enable us to find the premium on one dollar for any given age (without regarding the interest) : Divide the number of the deaths occurring between that age and one hun- dred by the number of premiums to be paid during the same time. In calculating the annual premiums at four per cent, interest the same general principles are applicable. In all these compu- tations it is found more convenient to discount the factors containing the number of persons living or dying rather than 16 rOHT OF [NRliBANCE. any given sum, as $1 or $1,000 to be insured, as the same result will be obtained. Ijet us refer to the first exa-mple given, in which the premium on a policy for $1,000, age ninety-nine, WHS $1,000, the insured being I'Cgarded as fcrtain to die by the end of the 'year. .\s all premiums are computed oji the assump- tion that the policies will have to be paid at the end of the year, the correct premium would be .Si, 000 discounted for one year a1 four per cent, inlorosl, or j.-ajX 1,000 = $9H].54. At the age ninety-eight, there being four persons living, the premiums paid would be represented as follows : PremiuiQS at the beginning of the first year 4. Premium second year discounted 961538 Total premiums 4.961538 At the end of the first year thei'e would be Three losses, present value =8 x .961538= 3.884614 End of second year, one loss=l x nn 4 x r-in= 925456 - .'^ L'^ Total present value of losses 3.810070 Dividing the discounted number of losses by the discounted number of premiums, and multiplying by $1,000, we have the annual premium for $1,000, age ninety-eight, (3.810070-=- 4.961538) X $1,000 = $'?67.92. The premium for the age ninety-seven is found in the same manner : 13x1= 13. 4 x T.k=4 X .961638= 3.846152 lxT.friXT.k=lx -935456= 924556 Discounted number of premiums 17.770708 9 X T.k=9 X .961538= 8.653843 3 XT.kxT.-k=3x. 934556= 3.773668 1 X T.ir X T.k X T.iT= Discounted number of losses 13.316506 Dividing the discounted number of losses by the discounted number of premiums, and multiplying by 1,000, we have (12.316506h-17.770708)x$1,000 = $693.19, the annual pre- mium at the age ninety-seven. The same method is employed COST OF INSURANCE. 17 in finding the annual premiums for all ages. The arithmetical computation of them would be exceedingly long, but by means of logarithms the process is comparatively easy. It will be noticed in the foregoing examples that the number of dying is multiplied by one more power of y-^ than the num- ber of the living. The reason of this is that the losses are regarded as taking place at the end of the year, while the premiums are paid at the commencement, and hence in finding the present value of the future losses we discount them by a power of y-Lj greater by one than is used in finding the present value of the future premiums. The method of finding the net annua] premiums b}^ means of the commutation columns is by dividing the discounted sura of the dying at any age in the M column by the discounted sum of the living in the N column, or ^-?^, and multiplying by 1,000. Thus, the net annual premium for a policy of $1,000 at the age twenty is (10722.8137^827931.05) x $1,000 = $12.95 ; that is, there would be 10722.8137 losses of $1,000 each to the payment of 827931.05 premiums, and consequently the net value of each preihium would be $12.95. The net single premium is the present value of the sum of all the net annual premiums on a given life, and is found by divid- ing the number in the M column at any given age by the corre- sponding number in the D column. Were it not for the element of discount which is employed in computing premiums, it is evident that the number of the living at any given age would be equal to the sum of the future deaths from that age. Thus, in Table I. the number of living at the age ninety is 1,319, which is equal to 427 + 322 + 231 + 155 + 95 + 52 + 24-f-9-^3 + l, the numb^ of dying between the ages ninety and one hundred, and the net single premium for $1,000 would be f|||x 1,000 = 1,000. But as the numbers of the djdng in the C column, whose sums make up the M column, are each discounted by a higher power of y Lj than the D number which we take for a denominator, the quotient is less than unity, and when multiplied by $1,000 is less than the sum insured. To find the net single premium for any given age, 2 18 COST OF INSURANCE. we divide the sum of all the discounted numbers of the dying by the discounted number of living at that age, and multiply the quotient by the sum insured. The formula is ^ x l,000=single premium for $1,000. Thus, the single premium for the age forty is ^"=(6242.4238-h16382.57) x $1,000 = $381.041. In a limited premium life policy, or one requiring a definite number of payments — ^as ten or twenty, for example — the premiums are graduated so that the sum of their present values equals the sum of the present values of the premiums on a continued premium policy of the same age. A man insuring at the age forty on the ten-annual life plan pays ten premiums, the present value of which at the age of issue is equal to the single premium, or to the present value of all the future annual premiums on a whole life policy issued at the same age. In the example of a whole life policy we found the annual premium by dividing M, the sum of the discounted numbers of the claims during life, by N, the sum of the discounted numbers of the future premiums. In a limited payment policy the same general rule is applicable, we find the annual premium by dividing the M number by the discounted number of premiums during the time when the premiums are payable, or by the difference between the N number at the age of issue and the N number at the age when the premiums cease. The premium 'on a ten annual life policy issued at the age ten is found, by referring to Table III., as follows : ^ lo— n To ^ $ljOOO = $1,000 X 14411. 3725h-(1381'771.'74 — 827931. 05) = $26.03. In term insurance the premiums are payable during a limited terra, of years, and the policy becomes a claim only in case the insured dies during that time. In this case the company merely carries the risk during the period of insurance. The premium is the ratio which the number of payments in the commutation column N, and during the specified time, bears j;o the number of deaths in column M during the same time multiplied by $1,000. The premium is found by dividing the difference between the M number when the policy commences and the M number of COST OF INSUEANCB. 19 the year when the policy ceases to be in force, by the difference between the corresponding N numbers. Referring to Table III., the net premium for a ten-annual term insurance issued at the age ten would be ^1^^^° x $1,000 = $1,000 x (14411.3725- 10722.8137) H- (1381771.74 — 827931.05) = (3688.5588 -4- 553840.69 x$l,000 = $G.66. Simple endowment is a form of insurance very seldom used, in which the policy is payable only in case the insured is alive at the time when the term of insurance expires. The annual premium is the ratio between the number of payments during this term and the probability that the party insured will be alive at the end of the term, and it is found by dividing the number of living in the D column at the time when the policy becomes payable by the difference between the N numbers at the beginning and end of the term of insurance. The annual premium for a simple endowment, issued at the age ten and payable at twenty if alive, is found by dividing (D 20 x $1,000) by (NIO-N 20) = ($1,000 x 42566.31) -^ (1381771.74- 827931. 05) = $76.86. Ordinary endowment insurance consists of two parts ; the term insurance and the simple endowment. We have explained how these two kinds of insurance are found separately, and it only remains to combine them to find the premiums on regular endowment insurance policies. To find the net premium for an endowment insurance age thirty-five, payable in fifteen years, the simple endowment part is found by dividing the discounted number of the living at the time when the payments cease, or at fifty years, by the discounted sums of the living between the ages specified. In the D column, opposite 50, we find the first numerator 9781.92, and the denominator is the dif- ference between the discounted sums of the living in N 35 and N 50, or 358785.65 — 131765.72=227019.93. The second part of the numerator, or the term insurance part, is found by taking the difference between the discounted sums of the dead in the M column at the ages thirty -five and fifty, or 7127.8675—4714.0142 = 2413.8533. Adding these two numerators together and >- 20i COST OF INSURANCE. ■=: II. IV. c:> L. c: r" • diwding by the denominator common to both, we have (9781.92 + ,24*3.8533) X $1,000-^227019. 93 = $5:i72, the premium re- Zqufred. Wai rhe general rule for finding the net premium for any rate is o siiiply this : Divide the number of claims, as represented in the p^bj the number of premiums during the payment of the same. Qo I| a single premium life policy the number of claims is repre- Q sJnted by M and the premiums by D ; in the annual premium life pfclicy the number of claims is also represented by M, and the imber of premiums is represented by N ; in the simple endow- |ent the claims are represented by D at the age when the ifisurance ceases, because the policy is payable only in case the flarty insured is alive at that time; and the number of premiums Ae denoted by the difference between N when the insurance commences and N when it ceases ; in term insurance the claims are denoted by the difference between M when the insurance commences and M when it ceases to be in force, and the num- ber of premiums are denoted by N in the same way. In a ten- annual life policy M represents the number of claims during life, and the number of payments during ten 3rears is denoted b}' N at the present age minus N at ten years older. All premiums, of whatever nature, are simply the ratios of the number of premiums to the number of claims, multiplied by the sum insured. VALUATION OF POLICIES. The meaning of the term value of a policy can be best under- stood by a practical illustration. The net premium on an ordinary life policy of $1,000, age forty, is 23.68. At the end of the first year this is increased by the standard rate of interest, four per cent., to 24.62. In the meantime the cost of carrying the risk is 10.21. Deducting this I'roni $24.62, and wo have a remainder at the end of the first year of 14.41. This is called by various names, as the "net valuo of the policy," -'reserve," "reinsurance fund," etc., and it is the unexpended or unearned* sum which must remain- on deposit in the hands of the com- COST OP INSURANCE. 21 pany, accumulating by annual additions and at compound interest, in order to meet the future claim which the policy- holder or his heirs will have upon the company. Each of the above appellations denotes the uses to which the value of a policy may b^ applied. The "net valae" signifies what the company can theoretically afford to return to the policy-holder provided he surrenders his policy; the "reserve" denotes the sum which the company must reserve or lay by to meet the future claim of the policy-holder; the "reinsurance fund" denotes the amount which the company would have to pay to transfer the policy, with the same terms upon which it was origi- nally issued, to another companj^. There are three distinct claims which- tbe gross premiums and interest on every policy are required to meet — the reserve, the cost of insurance, and the loading or the difference between the net and gross premiums. The loading is used to pay the expenses of the company, and the excess is applied to the dividends of the insured. Leaving out the loading from our calculations at present, we add the four per cent, interest to the net premium and deduct the cost of carrying the risk, and the remainder is the reserve. Another method of finding the reserve is founded on the definition of the value of a policy, which is the "difference between the present value of the sum insured and the present value of the premiums made to secure it." The net annual pre- mium in the foregoing example is 23.678. The present value of an annuity of $1, payable at the beginning of each year from the age of forty-one during life, is 15.861 ; hence, the present value of all the future premiums at this age would be 23. 6*78 x 15.861 = 375.55. The single premium at the age forty-one, or the present value of all the future net premiums at this age, is 389.96, and the difference between these is 14.41, the net value required. This rule may be expressed in the following proportion : As the single premium at the time when the policy is issued is to the present value of all the future premiums at the ,time when the policy is valued, so is $1,000 to the amount at risk on $1,000. By subtracting the amount at risk on $1,000 from $1,000, we have the value required. 22 COST OF INSURANCE. With the exception of those term insurance policies which run only for one year, the sum insured maybe divided into two parts: 1st, the amount at risk or the sum which the company Avould lose if the policy-holder should die during the year; and 2d, the value of the policy or self-insurance which the policy-holder assumes. If the policy-holder should die immediately after paying the first premium, the company would lose the difference between the net premium paid and the sum insured. As the company is at a continual expense during the year in carrying the risk, if the party insured should die at the end of the year, the company would lose the difference between the accumulated reserve and the sum insured, as the liability of the company continually increases from the beginning to the end of the insurance year. At the beginning of the second year the second premium is added to the reserve of the first year, when another gradual decrease in the reserve takes place, and con- tinues till the third premium is paid. It will be noticed that both the reserve and the amount at risk varies with considerable irregularity, the reserve increasing from year to year in a zigzag line, and is greatly modified by the interest on the reserve and the cost of insurance, which is still further modified by the age of the party insured. In short term endowments and policies which have but a few years to run the interest on the reserve sometimes becomes larger than the cost of carrying the risk, so that the value of the policy continually increases in an irregular manner. The amount at risk is the sum insured less the net value of the policy. At the age of forty the net, premium on a whole life policy of $1,000 is 23.68; hence, at the beginning of the first year the policy-holder insures himself in the amount of 23.68, and the company assumes a risk of 1,000 — 23.68 = 976.32. At the end of the first year this net premium has decreased to 14.41, and the compan}^ now carries a risk amount- ing to 1,000 — 14.41 = 985.59. At the beginning of the second year the second net premium is added to the reserve of the first year, 14.41-1-23.68 = 38.09, and the company now carries a risk of 1,000 — 38.09 = 961.91. At the end of the second year the reserve is diminished to 29.31, and the company carries a risk COST OF INSURANCE. 23 of 970.69. A.t the beginning of the third year the reserve is 29.31+23.68 = 52.99, and so on. From this explanation it will be seen that in case the policj'-holder should die during the time while the policy is in force, the company would lose, and for the same reason the heirs of the policy-holder would gain, not the- sum insured, but the difference between it and the self-insurance which the policy-holder assumes. The amount at risk varies greatly in different kinds of policies. Many people think that they get the same amount of insurance while paying the high premiums on a short term endowment as on an ordinary life policy, while the truth is that on a policy of a given amount the greater the premium the less insurance is purchased, and, on the other hand, the larger the premiums the less in number they are. In an ordinary ten-year endowment for $1,000, issued at forty, the reserve at the end of the first year is 79.66, and the amount at risk is 920.34. At the end of the second year the reserve is 163.16, and the amount at risk is 836.84. At the end of the third year the reserve is 250.72, and the amount at risk is 749.28. In this example the policy-- holder pays a net annual premium of 85.76, and gets less insurance than about one-quarter as much money would procure him if he adopted the whole life plan. There are advantages connected with both kinds of insurance. Short term endow- ments partake more of the savings banks principle, and should be patronized by those persons who are liable to meet with sudden reverses of fortune and be obliged to take paid-up policies for a proportional amount of the original sums insured. The cost of insurance is found by multiplying the amount at risk at any age by the quotient arising from dividing the number of dying by the number of living at that age, or, in other words, by the probability that the insured will die during that year. In term insurance for a single year, the cost of insurance is found by multiplying the whole amount insured by the probability of dying during that year. In this case there is no reserve required ; but in all cases where there is a reserve, the difference between it and the sum insured is made the basis of the calculation. 24 COST OF INSURANCE. Example. Life policy issued at age 40, for $1,000 00 Reserve at end of first year 14 41 Amount at risk $985 59 Number of dying, age 40 815 Number of living, age 40 78,65.S Probability of dying 815^78,653=. 010362 Cost of Insurance $986.59 x .010362=$10.21 Second year, policy §1,000 00 Reserve 29 31 Amount at risk $970 69 Number of dying, age 41 826 Number of living, age 41 77,838 Probability of dying 826-4-77,838=. 010612 Cost of insurance $970.69 x .010612=$10.30 The cost for subsequent years can be found in the same manner. A very easy way of finding the cost of insurance for the first year, when the reserves and net premiums have been obtained, is to add four per cent, interest to the net premium and subtract the reserve. For any subsequent year, the cost may be found by adding the net premium to the reserve of the previous year, increasing this amount by four per cent, interest, and subtracting the reserve for the given year, and the remain- der is the cost of insurance required. In the example already given, we have. First Year.— 'Set premium - ^23 68 Interest at four per cent 94 Amount ,«;24 62 Reserve 14 41 Cost of insurance |10 21 Second Trar.— Reserve first year $14 41 Net premium 23 68 Amount $38 09 Interest 4 per cent 1 52 $39 61 Reserve second year , 29 31 Cost of insurance |10 30 With the net premiums and tables of the values of policies, or the cost of insurance, nearly all the ordinary questions in life insurance can be easily solved. COST OF INSURANCE. 25 TABLE IV Net Rates — Four Per Cent. Annuities and Single Premiums. Present value of $1.00 per annum, to be received at the commencement Single Premium for whole Life Assur- o f every year during life. ance tor $1,000. Age. Age. Age, Age. $408.71 10 $20,454 48 $15,374 10 $313.32 43 11 20 869 44 15.119 11 216.56 44 418.52 12 20.282 45 14,857 12 219 98 45 438.57 13 20.191 46 14.590 13 328 44 46 488.86 14 20.096 47 14.817 14 227.08 47 449.85 15 19.998 48 14.039 15 230.86 48 460.02 16 19.896 49 13.757 16 234.78 49 470.88 17 19.790 50 18.470 17 238,84 50 481.91 18 19.681 51 18.179 18 243,05 51 493,11 19 19.567 52 12.884 19 247.40 52 504.46 20 19.450 58 12.585 20 251.91 53 515,95 21 19.880 54 12.283 21 356,56 54 527.57 22 19.204 55 11.978 22 361.38 55 589.81 23 19.075 56 11 670 28 366.86 56 551,16 24 18.9.41 57 11.359 34 271 50 57 563,10 2.5 18.803. 18.660* 58 11.046 35 276 82 58 575 14 36 59 10,731 26 • ^82. 81 59 587.36 27 18.512 60 10.415 27 287.99 60 599.43 28 18.360 61 10.098 28 298 86 61 611.63 29 18.202 63 9.780 29 299.91 62 628.88 30 18.040 68 9.464 30 806.17 68 686.00 81 17.872 64 9.149 31 312,62 64 648,13 32 17 698 65 8.835 33 319.29 65 660,17 83 17.520 66 8,525 83 336.17 66 672.13 34 17.885 67 8.217 34 338 37 67 683.96 85 17.144 68 7.913 85 H40.60 68 695.66 86 16.948 69 7.613 36 348,17 69 707.19 37 16.744 70 7.817 37 855.99 70 718,57 88 16.534 71 7.026 38 364.07 71 729,76 89 16.817 72 6.740 39 373.42 72 740 76 40 16.093 78 6.459 40 881.04 73 751.57 41 15.861 74 6.184 41 889,96 74 762.15 42 15.621 75 5.915 43 899.18 75 773.51 26 COST OF INSURANCE. DISTRIBUTION OF SURPLUS. In order to find the share of surplus in any year on a policy, according to the "Contribution" method, by the aid of these tables, it in necessary to know : 1st, the age of the party insured when the policy was issued ; 'id, the net premium ; 3d, the terms of the policy or the duration of the period of payment of premiums and the period of insurance ; 4th, the amount insured ; and 5th, the actual premium paid. There are three principal sources of surplus : 1st, the loading, or the difference between the actual and net premiums ; 2d, the difference between the assumed rate of interest on the assets, or four per cent., and the actual rate ; 3d, the favorable difference between the actual and tabular cost of insurance. The loading is designed to cover the expenses of the business and the possi- ble excess of the actual over the tabular mortality. When not exhausted by these two sources of expense, the remainder becomes surplus. The following examples, taken from the larger work on " The Cost of Insurance,'' heretofore referred to, will illustrate the method of finding the surplus on policies. Amounts insured, $1,000 each: Terms of Policy. Annual Premiums during Life. Age at issue. . Actual rates . . , Xet premiums Loading . . $26 38 19 87 $6 51 Ten- Annual Life. $52 40 42 06 $10 34 Twenty- i Ten- Year En- j Year En- dowment dowment. 85 S49 79 38 80 $10 99 $108 41 85 03 $28 38 It is assumed that interest is realized at the rate of six per cent, per annum, that the expenses are ton per cent, on the gross premiums, and that the actual cost of insurance is onlj^ three fourths of the tabular or expected cost as given in the tables. COST OF INSURANCE. 27 Loading Expenses Balance Interest six per cent . . • Surplus from Loading $6 51 $10 34 $10 99 2 64 5 24 4 98 3 87 . 5 10 6 01 23 31 36 4 10 5 41 6 87 10 84 12 54 75 13 29 The surplus from loading we assume to q,verage the same each year. The surplus from interest is found by taking the net premium, adding four per cent, interest, and deducting the cost of insur- ance, which gives the reserve. This reserve multiplied by the difference between the actual and assumed rate of interest (6 — 4 = 2) will give the surplus from the second source : Net premiums* Add four per cent, interest. Deduct cost first year Reserve end of first year. . . Add second net premium. . . Reserve outset second year. Interest Deduct cost second year. . . . Reserve end of second year Add third net premium Reserve outset third year . . . Interest Deduct cost third year Reserve end tliird year Add fourth net premium.. Reserve outset fourth year. . " fifth " . " sixth " . 87 79 66 08 58 87 35 25 60 26 34 87 21 73 94 34 60 87 47&C 13 21 $42 1 43 8 34 42 76 3 79 8 71 42 113 4 117 8 109 42 151 190 232 68 74 96 78 06 84 07 91 81 10 06 16 53 69 63 06 06 12&C 79 28 1 1 40 9 31 38 70 2 72 8 64 38 102 4 106 8 98 38 137 172 210 80 55 35 00 35 80 15 81 96 88 08 80 88 11 99 73 26 80 06&C 76 07 $85 8 79 85 164 6 171 7 163 85 248 9 258 7 351 85 336 428 524 03 40 43 55 88 03 91 60 51 93 58 03 61 94 55 25 30 03 33&C 30 78 Surplus from the second source, found by computing two per cent, interest on the reserves at the beginning of each year : First year. . Second year Third year. Fourth year Fifth year . . Sixth year. . $0 40 $0 84 $0 78 63 1 54 1 40 86 2 26 2 06 1 11 3 02 2 74 1 86 8 81 3 45 1 62 4 65 4 20 $1 70 3 30 4 97 6 73 8 56 10 50 * Eeserve at the outset of the first year. 28 COST OF INSURANCE. The surplus from the third source is found by taking one fourth of the cost of insurance, as given in the table : First year Second year | 2 Third year Fourth year Fiftli year Sixth year 2 29 $2 24 .^2 25 |2 13 2 31 2 20 2 22 , 1 98 a 33 2 16 2 18 1 81 2 36 2 11 3 14 ! 1 63 2 38 2 05 2 10 1 1 42 2 39 1 99 2 05 , 1 19 Adding the itenjs of surplus from these three sources, the amounts are : First year sO 79 Second year 7 04 Third year Fourth year Fiftli year Sixth year 7 29 7 57 7 84 8 11 $8 49 $9 40 9 15 9 99 9 83 10 61 10 54 11 25 11 27 11 92 12 05 12 62 $17 12 18 57 20 07 21 65 23 27 24 98 It will be seen from th'e results derived from the premises we have assumed, that the first year's dividend on the whole-life poliojr is 2G per cent.; on the ten-annual life, 16 percent.; on the twenty-year endowment, 1 9 per cent. ; and on the ten-year endowment, 16 per cent. -\t the sixth year these dividends are respectively, 31, 23, 25, and 23 per ceiit. SURRENDER VALUES. Under the head of " Yaluation of Policies," we explained the net value of a policy to be "what the compan}' can theoreticallv afford to return to the policy-holder provided he suri-enders his policy " and relieves the company fi'om any further obligation to him. If this rule were strictly followed out in practice it would often work great injustice to the remaining members. Some people who have an idea that their duration of life is short are desirous of obtaining the largest amount of insurance at the least immediate cost; there are others who having been insured COST OF INSURANCE. 29 several years find their vitality decreasing are very desirous of lieeping ttieir policies in force. It seldom happens that one of these two classes desires a surrender value of his policy. On - the other hand that class "whose confidence in their own vitality makes it comparatively a matter of indifference with them whether they keep themselves insured or not, are often the ones most desirous of giving up their policies and obtaining their full surrender value. It will be seen from this statement that it is practically impossible for a company to make an invariable rule on this subject which will operate impartially to all. In calculating the surrender value of a polic}^ there are two parties to be considered — the member who withdraws and those who remain insured. If all the healthy members, after having been insured a few years, were allowed to withdraw and receive in cash the net value of their policies, and all the dividends on them, an additional load of mortality would be thrown upon the remaining members, and they would be obliged to' assume a burden which they did not expect to bear when they entered the company. Their dividends would be diminished and the sol- vency of the company endangered. In dealing with the minority who withdraw, the superior claims of the great majority who remain should not be ignored. If a party requires a medical examination on entering a company, is it unreasonable that the state of his health should be taken into consideration when he withdraws, and should not the surrender value of his policy be graduated accordingly ? A policy of insurance is a definite con- tract on the part of the company to pay a certain amount on the death of the policj'-holder, or on his arriving at a certain age, in consideration of a premium to be paid by the policy-holder according to the terms of the policy. Unless there is a separate stipulation to pay a definite surrender value expressed in the policy, this is never a part of the original contract on the part of the company, and in demanding the surrender value the policy- holder usually asks the company to make a new contract entirely distinct from the original one. Whatever the company can actually afford to pay from the unearned part of the premiums 30 COST OF INSURANCE. received, in order to be relieved from any further obligation, and without infringing upon the rights of the remaining policy-holders, must depend upon the circumstances of each individual case. ASSURANCE VALUE OR REVERSIONARY INSURANCE. The dividends or surplus which has been described has an "assurance value," that is, it may be used to permanently in. crease the amount of the policy in the form of paid-up insurance- In this case the surplus may be considered as a single premium for an additional paid-up policy issued at the actual age of the insured, and payable when the original policy becomes a claim. Suppose, for example, that the surplus on an ordinary life policy of $10,000, issued at the age thirty-five, is $100 at the end of the third year, or when the insured reaches the age of thirty- eight years ; and the single premium on a whole life policy of $1,000, at the table rates of the company, and at the age thirty- eight is $425.64, the amount of paid-up insurance whicn $100 surplus will purchase will be found by the following proportion — viz., 425.64:100 : : 1,000:234.94. That is, as the single premium on $1,000 is to the surplus $100 regarded as a single premium, so is a paid-up policy of $1,000 to amount of rever- sionary insurance which $100 will purchase. This rule is also applicable to all life policies of a limited number of premiums. To find the amount of reversionary insurance on an endow- ment policy, the surplus is divided by the single premium of an endowment policy of $1 calculated for the remaining years dur- ing which the premiums are payable. REDUCTION OP PREMIUM. If the holder of an ordinary life policy chooses to have the surplus applied to the permanent reduction of the amount of pre- mium the amount of this reduction is found by dividing the sur- COST OF INSURANCE. 31 plus by "the present value of $1 per annum to be received at commencement of every year during life," according to the table of mortality and rate of interest adopted by the company. In the preceding examples the present value of $1 per annum, " Actuaries' Table, four per cent." (see Table IV,) at the age of thirty-eight is $16,534. According to this standard, $16,534 would purchase a permanent reduction of one dollar on an ordi- nary life policy at this age, and the amount which $100 will pur- chase is foundby the following proportion — 16.534 :1 : : 100 : 6.04, or $100 -H 16.534 = $6.04. It is evident from an inspection of the annuity columns in Table lY. that with a given amount of surplus the amount of reduction of premium would increase each successive year. The reduction of premium on a limited payment polic}', whether life or endowment, is. found by dividing the surplus by the present value of one dollar per annum, to be received at the commencement of each year during the remaining years in which the premiums are payable. The less the number of years the premiums are payable, and over which the surplus is to be dis- tributed, the larger, with a given amount of surplus, will be the reduction. PREMIUM AND COST OF INSURANCE TABLES. The following tables are based upon the Actuaries' or Com- bined Experience Table of Mortality, and four per cent, interest. They give the net premiums for nearly all kinds of policies in common use, and the cost of insurance for $1,000 during the first six years of the existence of a policy. 32 COST OF ITSrSUBANCE. TABLE V. Net Annual Premiums, Life— $1,000. Annual Annual dtjring Annual Annual ditking DT7BING Life. Ten Teaks. DUBING LiPE. Ten Yeabs 20 $12 95 $30 81 41 $24 59 $48 53 21 13 27 31 40 42 25 55 49 77 22 13 61 82 00 48 26 58 51 08 23 13 96 32 62 44 27 68 52 44 24 14 33 33 27 45 28 84 53 86 25 14 72 33 94 46 30 08 55 33 26 15 13 34 63 47 31 38 56 85 27 15 56 35 35 48 32 77 58 43 28 16 00 36 09 49 34 23 60 05 29 16 48 36 86 50 35 77 6174 ^ 30 16 97 37 66 ^ 51 37 41 63 49 31 17 49 88 48 52 39 15 65 30 32 18 04 39 38 58 41 00 67 17 33 18 62 40 21 54 42 95 69 12 34 19 22 41 12 55 45^02 71 14 35 19 87 42 00 56 47 23 73 25 36 20 54 43 04 57 49 57 75 44 37 21 26 44 05 58 52 07 77 74 38 22 02 45 10 59 54 72 80 15 39 22 82 46 19 60 57 56' 82 68 40 23 68 47 33 COST OF INSURANCE. 83 TABLE VI. ■ Net Annual Premiums, Endowment— 1,000. i Death Death Death Death Death Death Dbath Death < OE 35. ..OK 40. OK 45. , OK 60. ,qH 55. OK 60. OK 65. OK 70. 20 $52 27 $36 97 $28 19||22 68:$19 06 $16 64 $15 04 $14 02 21 56 79 39 41 29 71 23 70 19 79 17 20 15 49 14 41 22 62 02 42 15 31 36 24 80 20 58 17 80 15 97 14 81 23 68 14 45 21 33 18 25 99' 21 42 18 43 16 47 15 24 24 75 41 ,48 68 35 17 27 2,^ 22 32 19 10 17 01 15 70 25 84 15 52 62 37 38 28 68 23 29 19 82 17 58 16 17 26 94 86 57 14 39 84 30 21 24 33 20 £8 18 18 16 68 27 108 29 62 38 42 58 31 88' 25 45 21 40 18 81 17 21 28;i25 59 6S 52 45 66 33 71 26 67 22 27 19 4^1 17 n 29148 71 75 78 ,49 13 35 72 27 98 23 20 20 20 18 86 30181 11 84 53 -/53 08 37 95 29 40 24 21 20 97 18 99 31 95 26 57 62 40 42 30 96 25 28 21 78 19 65 32 108 69 62 86 43 17 32 65 26 44 22 65 20 85 33 125 99 69 00 46 26 34 50 27 70 •IB 57 21 10 34 149 10 76 28 49 75 36 55 29 05 24 57 ■21 89 35 181 50 85 03 53 72 38 80 30 52 25 63 22 74 36 95 76 58 27 41 30 32 1-J 26 78 23 64 37 109 19 63 53 44 09 33 87 28 01 24 60 38 126 49 69 69 47 22 35 78 29 34 25 63 39 149 59 76 99 50 75 ;57 89 30 78 26 73 40 181 98 85 76 54 77 40 21 32 34 27 92 41 96 51 59 38 42 80 34 05 29 19 42 109 98 64 71 45 68 35 91 30 57 48 127 33 70 95 48 91 37 96 32 06 44 150 48 78 33 52 56 40 20 33 67 45 182 92 87 21 56 70 42 68 85 42 4b i 98 05 61 48 45 41 37 81 47 111 60 66 87 4'S 44 39 35 48 129 01 78 22 51 81 41 58 49 152 21 80 70 65 59 44 00 50 184 66 89 6ti 59 86 46 65 51 100 58 64 72 49 57 52 ■ 114 19 70 29 52 78 53 . 131 66 76 76 56 35 54 154 89 84 37 60 33 55 187 34 93 45 64.80 56 104 48 69 87 57 118 20 75 65 58 135 75 82 34 59 159 04 90 17 60 191 49 99 47 3 34 COST OF INSURANCE. TABLE V]]. Net Annual Premiums, Endowment— $1,000. D OB 10 Yeabs. D OB 15 Yeabs. D OB 20 Years. D OB 25 Yeabs. D OE 30 Teabb. D OB 35 Yeabs. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 $83 86 83 91 83 97 84 03 84 09 84 15 84 22 84 29 84 37 84 45 84 54 84 63 84 72 84 82 84 92 85 03 85 15 85 28 85 42 85 58 85 76 85 98 86 22 86 51 86 84 87 21 87 62 88 06 88 55 89 08 89 66 90 29 90 98 91 73 92 55 93 45 94 43 95 52 96 71 98 02 99 47 $52 27 52 33 52 40 52 47 52 54 52 62 52 70 52 79 52 88 52 98 53 08 53 19 53 31 53 44 53 57 53 72 53 88 54 07 54 28 54 51 54 77' 55 07 55 41 55 79 56 22 1 56 70 57 23 57 80 58 43 59 11 59 80 60 68 61 58 62 56 63 63 64 80 66 09 67 51 69 06 70 77 1 72 64 $36 97 37 05 37 12 37 21 37 29 37 38 37 49 37 59 37 70 37 82 37 95 38 09 38 25 38 41 38 60 38 80 39 03 39 28 39 56 39 87 40 21 40 61 41 04 41 42 53 08 42 68 43 34 44 06 44 85 45 71 46 65 47 68 48 81 50 03 51 37 52 84 54 44 56 18 58 09 60 17 62 45 $28 19 28 28 28 37 28 47 28 57 28 69 28 81 28 94 29 08 29 24 29 40 29 59 29 79 30 01 30 25 30 52 30 (?2 31 14 31 50 31 90 32 35 32 84 33 39 34 00 34 68 35 42 36 23 37 12 38 08 39 13 40 27 41 51 42 87 44 33 45 93 47 66 49 53 i 51 57 53 78 56 17 58 76 $22 68 22 79 22 90 23 02 23 15 23 29 23 45 23 61 23 79 23 99 24 21 24 44 24 70 24 98 25 29 25 63 26 01 26 42 26 87 27 37 27 92 28 53 29 20 29 94 30 76 31 63 32 60 33 64 34 77 36 00 37 32 38 75 40 30 41 96 43 75 45 68 $19 06 19 19 19 33 19 48 19 64 19 82 20 01 20 22 20 45 20 70 20 97 21 26 21 59 21 94 22 32 22 74 23 20 23 70 24 25 24 85 25 50 26 22 27 01 27 87 28 80 29 82 30 91 32 09 33 35 34 71 36 16 37 72 39 39 41 18 43 08 45 12 COST OF INSURANCE. 85 TABLE VIII. Ordinary Life Policies-$1,000. FiBST Year. Second Yeah. Thied Year. FOTJBTH Yeak, Fifth Yeab. Sixth Xeab. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 ^4 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 61 52 63 54 65 56 57 58 59 60 $7 25 7 7 7 7 7 7 7 33 41 51 61 71 82 94 8 07 8 20 8 35 8 50 8 66 8 83 9 00 9 18 9 37 9 56 9 78 9 99 10 21 10 45 10 72 11 07 11 50 11 99 12 60 13 26 13 97 14 75 15 59 16 52 17 53 18 64 19 81 21 11 22 51 24 00 25 64 27 42- 29 41 28 37 46 56 66 76 8 00 8 13 8 27 8 41 8 57 8 74 8 90 9 08 9 26 9 45 9 65 9 86 10 07 10 30 10 56 10 89 11 30 11 79 12 37 13 01 13 70 14 46 15 27 16 16 17 14 18 20 19 33 20 60 21 93 23 36, 24 93 26 63 28 56 30 62 32 41 51 $7 37 17 41 10 16 10 41 10 72 11 13 11 59 12 16 12 77 13 44 14 17 14 96 15 83 16 77 17 80 18 88 20 09 21 39 22 76 24 26 25 90 27 73 29 72 31 92 I 7 60 7 71 7 82 7 94 8 06 8 20 8 34 8 49 8 65 8 81 8 98 9 16 9 34 9 54 9 74 9 94 46 55 66 76 7 7 7 7 7 88 8 00 8 13 8 27 8 42 8 67 8 73 8 89 9 07 9 24 9 43 9 62 9 81 10 03 10 27 10 57 10 96 11 41 11 96 12 65 13 20 13 90 14 67 15 61 16 42 17 41 18 46 19 62 20 87 22 19 23 64 25 21 26 97 28 88 30 99 33 24 7 7 7 7 7 8 50 60 71 82 94 07 8 20 8 35 8 49 8 64 8 81 8 97 9 14 9 33 9 51 9 70 9 90 10 13 10 43 10 80 11 23 11 77 12 34 12 97 13 65 14 40 16 21 16 09 17 06 18 06 19 18 20 40 21 66 23 06 24 56 26 26 28 08 30 09 32 26 34 63 17 46 7 66 7 65 7 77 7 88 8 01 8 14 8 28 8 42 8 67 8 72 8 89 9 06 9 23 9 40 9 58 9 78 10 00 10 29 10 65 11 11 07 59 12 16 12 76 13 42 14 14 14 92 16 78 16 71 17 69 18 77 19 93 21 16 22 60 23 95 26 68 27 33 29 26 31 4 33 61 36 08 36 COST OF INSURANCE. TABLE IX. Ten-Premium Life Policies. Fib ST Second Thibd FonETH Fifth Sixth Ag-k Yeae. Year. Teae. Yeae. Yeae. Yeae. 20 $7 11 $7 00 $6 88 $6 76 $6 62 $6 47 21 7 19 7 08 6 96 6 83 6 69 6 54 22 7 27 7 16 7 04 6 91 6 77 6 61 23 7 36 7 25 7 12 6 99 6 84 6 69 24 7 46 7 34 7 22 7 08 6 93 6 77 25 7 56 7 44 7 31 7 17 7 02 6 86 26 7 66 7 54 7 42 7 27 7 12 6 95 27 7 77 7 66 7 52 7 38 7 22 7 05 28 7 90 7 77 7 64 7 49 7 33 7 15 29 8 03 7 90 7 76 7 61 7 44 7 25 30 8 16 8 04 7 90 7 74 7 55 7 36 31 8 31 8 18 8 03 7 86 7 68 7 47 32 8 46 8 33 8 17 8 00 7 80 7 58 33 8 62 8 48 8 32 8 14 7 98 7 70 34 8 79 8 64 8 47 8 28 8 07 7 82 35 8 96 8 81 8 63 8 43 8 20 7 94 36 9 15 8 98 8 80 8 59 8 34 8 07 37 9 38 9 17 8 97 8 74' 8 49 8 22 38 9 54 9 36 9 15 8 92 8 67 8 43 39 9 74 9 55 9 34 9 11 8 90 8 69 40 9 95 9 76 9 55 9 36 9 19 8 99 41 10 19 10 00 9 83 9 68 9 53 9 38 42 10 45 10 30 10 19 10 06 9 95 9 79 43 10 78 10 69 10 60 10 52 10 41 10 23 44 11 19 11 13 11 10 11 03 10 90 10 72 45 11 67 11 68 11 65 11 57 11 44 11 24 46 12 27 12 27 12 24 12 16 12 02 11 81 47 12 90 12 91 12 89 12 81 12 66 12 43 48. 13 59 13 61 • 13 59 13 51 13 36 13 10 49 14 34 14 38 14 36 14 28 14 11 13 82 50 15 16 15 21 15 20 15 12 14 91 14 60 51 16 05 16 12 16 12 16 01 15 80 15 44 52 17 04 17 11 17 09 16 99 16 75 16 33 53 18 11 18 17 18 17 18 05 17 76 17 30 54 19 25 19 34 19 33 19 17 18 86 18 35 55 20 51 20 61 20 56 20 40 20 05 19 53 56 21 87 21 95 21 92 21 73 21 40 20 81 57 23 32 23 42 23 39 23 24 22 86 22 22 58 24 91 25 03 25 05 24 87 24 47 23 75 59 26 <)5 26 83 26 85 26 68 26 22 25 42 60 28 60 28 80 28 84 28 64 28 14 27 24 COST OF INSURANCE. 37 TABLE X. Single Premium Life Policies. Age. Cost of iNStlEANOK. AGE. Cost op Instjranok. Age. Cost of Insurance. Age. Cost of Insukancb. 20 $5 42 31 $5 84 42 $6 44 53 $9 02 21 5 45 82 5 89 48 6 54 54 9 36 22 5 48 33 5 95 44 6 68 55 9 72 23 5 51 34 6 00 45 6 85 56 10 11 24 5 54 85 6 05 46 7 07 57 10 49 25 5 58 36 6 11 47 7 30 58 10 89 26 5 62 37 6 16 48 7 54 59 11 32 27 5 65 38 6 22 49 7 80 60 11 78 28 5 70 39 6 27 50 8 08 29 5 74 40 6 32 51 8 37 30 5 79 1 41 6 38 52 8 69 Endowment, Death or 35, Age. FiSST SeOONI) Thibd POURTH Fifth Sixth Yeae. Year. Ybak. Ybak. Year. Year. 20 $6 95 $6 66 $6 35 $6 03 $5 67 $5 29 21 6 99 6 67 6 33 5 96 5 55 5 12 22 7 04 6 68 6 28 5 86 5 40 4 90 23 7 08 6 67 6 21 5 73 5 20 4 64 24 7 12 6 64 6 12 5 56 4 95 4 29 25 7 15 6 59 5 98 5 33 4 62 3 85 Endowment. Death or 40. First Yeau Second Yeab. Thibd Yeab. FOTTRTH Yeab. Fifth Yeab. Sixth Yeab. 20 21 22 23 24 25 26 27 28 29 30 $7 06 7 13 7 19 7 26 7 33 7 40 7 48 7 55 7 62 7 69 7 75 $6 90 6 95 00 04 08 12 16 18 20 20 17 m 73 6 76 79 80 6 82 6 82 6 81 6 78 6 74 6 66 6 54 $6 55 6 56 6 55 6 54 $6 35 6 33 6 30 6 26 6 52 6 21 6 49 6 18 6 43 6 02 6 35 5 88 6 24 5 69 6 08 5 43 5 85 5 09 $6 13 6 09 6 03 5 96 5 86 5 5 5 5 4 4 74 57 36 09 72 25 38 COST OF INSURANCE. TABLE XI. Endowment. Death or 45. Age. First Second Tbibd Fourth Fifth Sixth Yeae. Teae. Yeah. Yeak. Yeae. Te»k. 20 $7 13 87 04 $6 94 $6 85 $6 74 $6 62 21 7 20 7 10 7 00 6 89 6 77 6 63' 22 7 28 7 17 7 06 6 93 6 79 6 65 23 7 36 7 24 7 11 6 97 6 82 6 66 24 7 44 7 31 7 17 7 01 6 84 6 66 25 7 53 7 38 7 22 7 05 6 86 6 65 26 7 62 7 45 7 28 7 08 6 87 6 63 27 7 71 7 53 7 38 7 11 6 86 6 60 28 7 82 7 60 7 38 7 12 6 85 6 54 29 7 92 7 68 7 42 7 13 6 81 6 45 30 8 03 7 75 7 45 7 12 6 74 6 34 31 8 14 7 82 7 47 7 08 6 65 6 17 32 8 25 7 88 7 46 7 01 6 51 5 95 33 8 36 7 92 7 44 6 90 6 31 5 66 34 8 45 7 94 7 37 6 74 6 05 5 27 35 8 55 7 93 7 25 6 51 5 68 4 76 Endowment. Death or 50. First Second Third FOUBTH Fifth Sixth Year. Year Year. Year Teae. Yeae. 20 $7 17 $7 13 $7 08 $7 03 16 98 $6 92 21 7 25 7 20 7 15 7 10 7 03 6 97 22 7 33 7 28 7 22 7 16 7 09 7 02 23 7 42 7 36 7 29 7 23 7 15 7 07 24 7 51 7 44 7 37 7 29 7 21 7 12 25 7 60 7 53 7 45 7 37 7 27 7 17 26 7 70 7 62 7 53 7 44 7 34 7 22 27 7 80 7 72 7 62 7 51 7 39 7 26 28 7 92 7 82 7 71 7 59 7 45 7 30 29 8 04 7 92 7 80 7 66 7 50 7 32 30 8 16 8 03 7 89 7 73 7 54 7 34 31 8 29 8 14 7 98 7 78 7 57 7 34 32 8 43 8 25 8 06 7 84 7 59 7 31 33 8 57 8 36 8 14 7 88 7 59 7 27 34 8 70 8 47 8 20 7 90 7 57 7 19 35 8 85 8 57 8 26 7 91 7 52 7 08 36 9 00 8 66 8 30 7 89 7 43 6 91 37 9 14 8 75 8 32 7 83 7 29 6 70 38 9 28 8 82 S 30 7 73 7 11 6 44 39 9 41 8 86 8 25 7 59 6 88 6 10 40 9 54 8 88 8 16 7 40 6 56 5 62 COST OF INSURANCE. 89 TABLE XiJ. Endowment. Death or 55. First Sfcond Third Fourth FrPTH Sixth AGE. Year. Year. Year Year. Year. Year. 20 $7 20 $7 19 $7 17 $7 16 $7 14 f7 12 21 7 28 7 26 7 25 7 23 7 21 7 19 22 7 36 7 35 7 33 7 31 7 29 7 26 23 7 45 7 43 7 41 7 39 7 36 7 34 24 7 55 7 52 7 50 7 47 7 45 7 41 25 7 64 7 62 7 59 7 56 7 53 7 49 26 7 75 7 72 7 69 7 66 7 62 7 58 27 7 86 7 83 7 79 7 76 7 71 7 66 28 7 98 7 94 7 90 7 86 7 81 7 75 29 8 10 8 06 8 02 7 96 7 90 7 82 30 8 24 8 19 8 14 8 07 7 99 7 90 31 8 38 8 32 8 26 8 17 8 08 7 98 32 8 52 8 46 8 37 • 8 28 8 17 8 04 33 8 68 8 59 8 50 8 38 8 25 8 10 34 8 83 8 73 8 62 8 48 8 33 8 15 35 9 00 8 88 8 73 8 58 8 40 8 18 36 9 17 9 02 8 86 8 67 8 45 8 20 37 9 33 9 17 8 97 8 74 8 49 8 22 38 9 52 9 31 9 08 8 81 8 53 8 25 39 9 69 9 45 9 18 8 88 8 59 8 30 40 9 87 9 59 9 28 8 98 8 68 8 34 41 10 07 9 74 9 43 9 11 8 75 8 37 42 10 28 9 94 9 60 9 23 8 83 8 33 43 10 54 10 19 9 79 9 36 8 83 8 18 44 10 87 10 45 9 99 9 43 8 73 7 89 45 11 24 10 75 10 14 9 40 8 48 7 37 46 11 68 11 02 10 21 9 22 8 01 6 54 47 12 11 11 22 10 13 8 80 7 19 5 23 48 12 53 13 31 9 82 8 02 5 84 3 20 49 12 87 11 18 9 13 6 65 3 64 00 50 13 09 10 69 7 78 4 26 00 40 COST OF INSURANCE. TABLE XIII. Endowment. Death or 60. Agk. riBST Second Thied Fourth Fifth Sixth Year. Yeab. Year. Yeab. Teae. Yeab. 20 $7 22 $7 23 $7 23 $7 24 $7 25 $7 25 21 7 30 7 30~ 7 32 7 32 7 33 7 33 22 7 38 7 39 7 40 7 40 7 41 7 42 23 7 48 7 48 7 49 7 50 7 50 7 51 24 7 57 7 58 7 58 7 59 7 60 7 60 25 7 67 7 68 7 68 7 69 7 69 7 70 26 7 78 7 78 7 79 7 80 7 80 7 80 27 7' 89 7 90 7 90 7 91 7 91 7 91 28 8 02 8 02 8 03 8 03 8 03 8 02 29 8 14 8 15 8 15 8 15 8 15 8 13 30 8 28 8 28 8 29 8 28 8 26 8 25 31 8 43 -8 43 8 42 8 41 8 39 8 36 32 8 58 8 58 8 56 8 54 8 51 8 47 33 8 74 8 72 8 71 8 68 8 63 8 59 34 8 90 8 88 8 85 8 81 8 76 8 69 35 9 08 9 05 9 00 8 95 8 88 8 80 36 9 26 9 21 9 16 9 09 9 00 8 90 37 9 44 9 39 9 31 9 22 9 12 9 02 38 9 63 9 56 9 47 9 36 9 26 9 18 39 , 9 83 9 74 9 63 9 52 9 44 9 39 40 10 04 9 92 9 81 9 73 9 67 9 62 41 10 25 10 13 10 04 9 99 9 94 9 90 42 10 49 10 40 10 35 10 29 10 26 10 19 43 10 80 10 74' 10 68 10 65 10 58 10 46 44 11 19 11 13 11 10 11 02 10 89 10 70 45 11 64 11 60 11 52 11 39 11 19 10 91 46 12 18 12 09 11 95 11 74 11 45 11 06 47 12 75 12 61 12 39 12 08 11 66 11 11 48 13 36 13 13 12 80 12 36 11 78 11 03 49" 14 01 13 66 13 19 12 57 11 77 10 72 50 14 69 14 18 13 52 12 65 11 53 10 12 51 15 39 14 67 13 73 12 51 10 98 9 06 52 16 11 15 07 13 73 12 06 9 94 7 30 53 16:80 15 30 13 43 11 08 8 13 4 49 54 17 39 15 27 12 59 9 24 5 10 00 55 17 .83 14 71 10 79 5 96 00 COST OF INSURANCE. 41 TABLE XIV. Endowment, Death or 65. Age. FiBST Second Third FOUBTH Fifth .Sixth Year. Yeab. Yeab. Yeak. Yeak. Year. 20 $7 23 $7 25 $7 27 $7 30 $7 82 $7 34 21 7 31 7 33 7 36 7 38 7 40 7 48 22 7 40 7 42 7 45 7 47 7 49 7 52 23 7 49 7 52 7 54 7 57 7 59 7 62 24 7 59 . 7 61 7 64 7 66 7 69 7 72 25 7 69 7 72 7 74 7 77 7 80 7 84 26 7 80 7 83 7 86 7 89 7 92 7 95 27 7 91 7 95 7 98 8 01 8 04 8 08 28 8 04 8 07 8 10 8 14 8 17 8 20 29 8 17 8 20 8 24 8 27 8 30 8 88 30 8 31 8 84 8 38 8 41 8 43 8 46 31 8 46 8 49 8 58 8 55 8 58 8 60 82 8 62 8 65 8 67 8 70 8 72 8 73 38 8 78 8 80 8 88 8 85 8 86 8 88 34 8 95 8 97 8 99 9 01 U 02 9 02 35 9 12 9 15 9 16 9 17 9 17 9 16 36 9 31 9 32 9 34 9 33 9 32 9 31 37 9 50 9 51 9 51 9 50 9 48 9 48 38 9 70 9 70 9 69 9 67 9 67 y 70 39 9 91 9 90 9 88 9 87 9 90 9 98 40 10 12 10 10 10 10 10 18 10 21 10 31 41 10 '35 10 34 10 37 10 46 10 56 10 71 42 10 60 10 64 10 72 10 83 10 99 11 14 48 10 93 11 02 11 13 11 29 11 45 11 59 44 11 84 11 45 11 62 11 78 11 93 12 05 45 11 82 U 99 12 15 12 30 12 43 12 54 46 12 39 12 56 12 71 12 85 12 96 13 03 47 13 01 13 17 13 31 13 43 18 51 18 54 48 18 69 18 83 13 95 14 03 14 06 14 03 49 14 41 14 58 14 61 14 65 14 62 14 48 50 15 19 15 28 15 31 15 28 15 14 14 90 51 16 03 16 07 16 04 15 89 15 64 15 23 52 16 94 16 91 16 75 16 48 16 06 15 41 53 17 91 17 74 17 46 17 01 16 33 15 39 54 18 92 18 62 18 13 17 40 16 41 15 09 55 19 99 19 47 18 69 17 63 16 20 14 37 42 COST OF INSURANCE. TABLE X V. Endowment. Death or 70. Age FlEST Second Thiud 3?OOKTH Fifth Sixth 20 Year. Yeah. > EAR. Yeae. Ykak. Yeab. $7 24 $7 27 $7 29 $7 33 $7 37 $7 40 21 7 32 7 35 ; 7 38 7 42 7 45 7 49 22 7 40 7 44 7 48 7 51 . 7 55 7 59 23 7 50 7 54 7 57 7 61 7 65 7 69 24 7 60 7 63 7 67 7 71 7 76 7 80 25 7 70 7 74 7 78 7 82 7 87 7 92 26 7 81 7 85 7 90 7 94 7 99 8 04 27 7 93 7 97 8 02 8 07 8 12 8 18 28 8 05 8 10 8 15 8 20 8 26 8 31 29 8 19 8 24 8 29 8 34 8 40 8 44 30 8 33 8 38 8 44 '' 8 49 ' 8 54 8 59 31 8 4h ; 8 53 i 8 59 1 8 64 1 8 69 8 74 32 8 64, 8 69 ' 8 74 8 80 8 85 8 89 33 8 80 8 85 8 91 8 96 : 9 00 9 05 34 8 97 9 03 9 08 9 12 ; 9 17 9 21 35 9 15 ■ 9 21 . 9 25 9 30 ' 9 34 9 37 36 9, 34 9 39 9 44 9 48 , 9 51 9 55 37 9 53 9 58 9 62 9 66 9 69 ' 9 74 38 9 74 9 78 9 82 9 85 9 90 10 00 39 9 95 9 98 10 02 10 07 ; 10 17 10 32 40 10 17 10 21 10 25 10 36 10 51 10 70 41 10 40 10 45 10 55 10 71 10 91 11 16 42 10 67 10 77 10 93 11 14 11 39 , 11 ()6 43 11 00 11 17 11 37 11 64 11 91 12 19 44 11 42 11 63 11 90 12 18 12 48 i 12 77 45 11 91 12 19 12 48 12 78 13 08 13 38 46 12 50 12 80 13 10 13 41 13 72 i 14 03 47 13 14 13 45 13 77 14 09 14 41 14 73 48 13 84 14 16 14 49 14 82 15 15 15 47 49 14 59 14 93 15 27 15 61 15 94 16 22 50 15 41 15 76 16 11 16 45 ; 16 74 17 02 51 16 30 16 66 17 02 17 32 [ 17 61 17 84 52 17 27 17 63 17 95 18 25 18 49 18 64 53 18 33 18 65 18 96 19 21 j 19 37 : 19 45 54 19 43 19 76 20 02 20 18 ' 20 27 20 23 55 20 65 20 93 21 10 21 18 21 15 21 00 56 21 95 22 13 22 22 22 19 22 03 21 66 57 23 31 23 41 23 37 23 21 22 82 22 17 58 24 78 24 74 24 57 24 15 23 47 22 89 59 26 34 26 16 25 72 24 99 23 84 22 19 60 28 05 27 58 26 80 25 56 23 80 21 34 COST 0¥ INSURANCE. 43 / TABLE XVI. Ten-Year Endowment. First Second ! Third FODliTU Fifth Sixth ?0 Yeae. Yeah. Veak. Yeae. Yeak. Year. 86 70 16 16 '' $5 58 $4 95 $4 28 $3 55 ■21 6 78 ; 6 24 5 65 5 02 4 34 3 60 22 6 86 6 32 5 73 5 09 ' 4 40 3 66 23 6 96 6 40 5 81 5 17 4 47 3 72 24 1 7 05 6 49 ' 5 89 5 24 4 54 3 78 25 7 15 6 59 5 98 5 33 4 62 3 85 26 7 25 6 69 ■ 6 08 5 42 4 71 3 92 27 7 36 1 6 80 6 19 5 52 4 79 4 00 28 7 49 6 92 6 30 5 63 4 89 4 08 29 7 61 7 04 6 42 5 74 4 99 4 16 ao 7 75 7 17 6 54 5 85 5 09 4 25 31 7 89 7 31 6 67 5 97 5 20 4 35 32 8 05 7 46 6 81 6 09 5 31 4 44 33 8 21 7 61 6 95 6 23 5 42 4 54 34 8 37 7 77 7 10 6 36 5 55 4 65 35 8 55 7 93 7 25 6 51 5 68 4 76 36 8 73 8 10 7 42 6 66 5 81 4 87 37 8 91 «• 29 7 59 6 81 5 95 5 01 . 38 9 12 8 18 7 76 6 97 6 11 5 17 39 9 32 8 67 7 95 7 16 6 31 5 38 40 9 54 8 88 8 16 7 40 6 56 5 62 41 9 77 9 12 8 43 7 69 6 86 5 91 42 10 03 9 42 8 77 8 03 7 21 6 23 43 10 36 i> 79 9 16 8 45 7 60 6 58 44 10 77 10 22 9 63 8 90 8 02 6 96 45 11 24 10 75 10 14 9 40 8 48 7 37 46 11 82 11 32 10 71 9 94 8 99 7 83 47 12 45 11 95 11 32 10 52 9 54 8 33 48 13 13 12 63 11 98 11 17 10 15 8 88 49 13 88 13 37 12 72 11 88 10 81 9 46 50 14 611 14 18 13 52 12 65 11 53 10 12 51 15 58 15 07 14 39 13 48 12 31 10 83 52 16 55 16 04 15 33 14 40 13 17 11 59 53 17 61 17 08 16 37 15 39 14 09 12 42 54 18 74 18 23 17 49 16 45 15 10 13 34 55 19 99 19 47 18 69 17 62 16 20 14 37 56 21 34 20 80 20 00 18 90 17 45 15 51 57 22 79 22 25 21 44 20 34 18 81 16 76 58 24 38 23 84 23 06 21 92 20 32 18 14 59 26 10 25 63 24 83 23 66 21 97 19 66 60 28 05 27 58 26 79 25 56 23 80 21 3'4 44 COST OF INSURANCE. T VHLE X^ll. Fifteen-Year Endowment, Age. First Year. Second Year. Third Year. Fourth Year. Fifth Year. Sixth Year. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 $»i 95 03 7 7 7 7 7 7 7 11 21 30 40 51 63 76 7 88 8 03 8 i: 8 34 8 50 8 67 8 85 9 04 9 23 9 44 9 66 9 87 10 11 10 38 10 72 11 15 11 o3 12 23 12 88 66 74 83' 92 02 12 23 35 48 61 75 90 8 06 8 22 8 39 8 57 8 7»i 8 95 9 16 9 37 9 59 9 82 10 16 10 57 6 6 6 7 7 7 7 7 7 7 7 11 11 03 60 13 59 13 61 14 35 14 41 15 19 15 28 ]6 10 16 23 17 11 17 27 18 20 18 38 19 36 19 60 20 65 20 93 22 05 22 34 23 53 23 88 25 15 25 57 26 93 27 46 28 99 29 53 12 21 12 89 35 44 6 53 6 61 6 71 6 82 6 93 7 05 7 18 31 45 60 75 92 8 08 8 26 8 44 8 64 8 83 9 04 9 28 9 59 9 96 10 40 10 94 11 52 12 15 12 84 13 59 14 42 15 31 16 30 17 35 18 51 19 76 21 10 22 56 24 16 25 95 27 89 30 07 $6 03 6 11 19 29 38 6 49 6 60 6 72 6 84 6 98 12 26 41 57 73 91 8 09 8 27 •^ 47 8 70 8 98 9 34 !i 75 10 25 10 79 11 39 12 03 1-J 74 13 51 14 36 15 28 16 27 17 36 18 54 19 80 21 18 22 69 24 38 26 23 28 26 30 48 $5 67 5 75 5 84 5 93 6 03 6 13 6 24 6 36 6 48 6 61 6 74 6 89 7 7 7 7 7 7 88 8 08 8 35 8 68 9 03 19 35 52 69 06 9 63 10 03 10 59 11 19 11 85 12 57 13 36 U 22 15 14 16 16 17 27 18 44 19 74 21 15 22 73 24 46 26 37 28 45 30 73 $5 29 5 37 45 54 64 74 84 96 6 08 6 20 6 34 6 47 6 61 6 76 6 92 08 25 44 68 99 8 34 8 77 9 23 9 75 10 30 10 91 11 58 12 31 13 10 13 96 14 90 15 93 17 02 18 22 19 53 21 00 22 61 24 38 26 32 28 44 30 78 COST OF INSURANCE. 45 TABLE XVni. Twenty-Year Endowment First Year Second Year. Third Year. Fourth Year. Fii«rii Year. Sixth Year. 20 21 ■ 22 23 24 25 26 27 28 29 30 81 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 06 15 23 33 43 53 64 76 88 8 02 8 16 8 31 8 47 8 64 8 81 9 00 9 19 9 38 9 59 9 81 10 03 10 27 10 54 10 89 11 32 11 81 12 42 13 07 13 79 14 56 15 41 16 34 17 35 18 45 19 63 20 93 22 33 23 83 25 46 27 25 29 25 16 90 08 17 27 38 49 62 75 8 03 8 19 8 35 8 51 8 69 8 88 9 07 9 27 9 48 9 69 9 92 10 18 10 51 10 93 10 41 11 99 12 62 13 31 14 06 14 87 15 76 16 74 17 80 18 93 20 18 21 54 22 97 24 54 26 26 28 18 30 27 $6 73 6 82 6 91 7 01 7 11 7 22 7 34 7 46 7 60 7 74 7 89 8 04 8 20 8 38 8 55 8 74 8 93 9 13 9 34 9 56 9 81 10 13 10 52 10 98 11 54 12 15 12 82 13 53 14 32 15 18 16 11 17 13 18 22 19 42 20 73 22 10 23 61 25 26 27 10 29 11 31 32 $6 55 6 64 6 73 6 83 6 93 05 17 30 43 58 73 8 05 8 22 8 39 8 58 8 77 8 97 9 18 9 42 9 73 10 11 10 55 11 08 11 66 12 30 12 99 13 74 14 67 15 47 16 45 17 49 18 65 19 21 90 22 22 67 24 25 26 01 27 94 30 05 32 34 $6 35 6 44 6 53 6 63 6 74 6 86 6 98 11 25 39 54 70 86 8 03 8 21 8 40 8 59 8 79 9 02 9 31 9 67 10 09 10 60 11 16 11 77 12 43 13 15 13 94 14 80 15 74 16 74 17 85 19 05 20 31 21 70 23 21 24 90 26 75 28 77 30 97 33 37 $6 13 6 22 6 32 6 42 6 53 6 65 6 77 6 91 04 18 34 49 65 83 00 18 38 60 87 22 62 10 11 10 64 n 22 11 85 12 54 13 29 14 11 15 01 15 97 17 02 18 16 19 38 20 70 22 15 23 77 25 53 27 46 29 56 31 86 34 36 46 COST OF INSURANCE. TABLE XIX. Twenty-five-Year Endowment, Age 20 21 22 2.3 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 First Year. 13 21 30 40 50 60 71 83 96 8 09 8 24 8 39 8 55 8 72 8 89 9 08 9 27 9 47 9 68 9 90 10 12 10 36 10 63 10 98 11 41 11 91 12 51 13 17 13 89 14 67 16 52 16 45 17 46 18 57 19 74 21 05 22 45 23 95 25 59 27 37 29 37 Second Yeau. 04 13 22 32 42 53 64 77 90 04 19 35 51 68 9 05 9 24 9 44 9 65 9 87 ]0 10 10 87 10 70 11 12 11 60 12 19 12 83 13 53 14 28 15 10 16 00 16 98 18 05 19 19 20 44 21 80 23 24 24 81 26 53 28 45 80 54 Third Year. $6 95 04 13 23 34 45 57 70 84 9« 8 14 8 30 8 46 8 64 8 82 9 00 9 20 9 41 9 61 9 84 10 09 10 42 10 82 11 29 11 86 12 48 13 15 13 88 14 68 15 55 16 50 17 51 18 63 19 84 21 15 22 54 24 06 25 70 27 55 29 56 31 77 Fourth Ykae, |6 85 6 94 03 14 25' 37 49 63 76 92 8 07 8 23 8 40 8 58 8 76 8 95 9 15 9 35 9 57 9 82 10 13 10 52 10 97 11 52 12 12 12 78 13 48 14 25 15 10 16 01 17 01 18 08 19 25 20 51 21 85 23 31 24 90 26 68 25 61 30 73 33 02 Fifth Ykae, m 74 6 83 6 93 04 15 27 40 54 69 84 7 7 7 7 7 7 7 7 99 8 16 8 33 8 50 8 69 8 88 9 08 9 29 9 53 9 83 10 21 10 64 11 18 11 76 12 39 13 07 13 82 14 63 15 52 16 49 17 52 18 65 19 87 21 22 24 11 25 82 27 68 29 72 31 92 34 33 16 57 Sixth Yeab. 61 71 82 6 93 7 7 7 7 7 7 7 04 17 30 45 59 74 90 8 07 8 24 8 42 8 61 8 80 9 00 9 23 9 52 9 89 10 31 10 83 11 38 J2 00 12 66 13 38 14 17 15 02 15 96 16 95 18 05 19 23 20 47 21 84 23 81 24 97 26 75 28 72 30 84 33 15 35 66 t COSl OF INSUBANCE. 47 TABLE XX. Thirty -Year Endowment, Age. First Second Third Fourth Fifth Sixth. Yka«. Yeab Year. Year. Year. Year. 20 17 17 17 13 $7 08 $7 04 $6 98 $6 92 21 7 26 7 21 7 18 7 13 7 07 7 02 22 7 34 7 31 7 27 7 23 7 18 7 13 23 7 44 7 41 7 37 7 38 7 29 7 25 24 7 54 7 51 7 48 7 44 7 41 7 36 25 7 64 7 62 7 59 7 56 7 53 7 50 26 7 76 7 73 7 72 7 69 7 66 7 63 27 7 87 7 86 7 84 7 83 7 80 7 78 28 8 00 7 99 7 98 7 97 7 95 7 93 29 8 14 8 14 8 13 8 12 8 11 8 08 30 8 -^S 8 28 8 29 8 28 8 26 8 24 31 8 43 8 44 8 45 8 44 8 43 8 42 82 8 60 8 61 8 61 8 61 8 61 8 59 33 8 77 8 78 8 79 8 79 8 78 8 78 34 8 94 • 8 96 8 97 8 97 8 98 8 97 35 9 13 9 15 9 16 9 17 9 17 9 16 36 9 32 9 34 9 86 9 37 9 87 9 87 87 9 51 9 54 9 56 9 57 9 58 9 60 38 9 73 9 76 9 77 9 79 9 82 9 90 39 9 94 9 97 10 00 10 04 10 13 10 27 40 10 17 10 20 10 25 10 35 10 51 10 70 41 10 41 10 47 10 58 10 76 10 95 11 23 42 10 68 10 80 10 98 11 20 11 49 11 79 43 11 02 11 22 11 45 11 76 12 08 12 42 44 11 46 11 70 12 03 12 36 12 72 13 08 45 11 96 12 30 12 65 13 01 13 41 13 82 46 12 57 12 98 13 3a 13 73 14 16 14 61 47 13 22 18 63 14 06 14 51 14 98 15 48 48 13 94 14 39 14 86 15 35 15 88 16 42 49 14 72 15 21 15 73 16 27 16*85 17 42 50 15 57 16 11 16 68 17 27 17 87 18 52 51 16 50 17 09 17 71 18 34 19 01 19 70 52 17 51 18 16 18 81 19 51 20 23 20 95 53 18 62 19 29 20 02 20 77 21 52 22 26 54 19 79 20 54 21 32 22 10 22 92 23 78 • 55 21 09 21 90 22 70 23 56 24 45 25 42 • I 48 COST OF INSURANCE. TABLE XXI. Thirty- ive-Year Endowment. Age. First Seoond Third POUETH Fifth Sixth Yeae Yeae. Teak. Teae. Yeab. Year. 20 $7 20 $7 19 $7 17 $7 16 $7 14 $7 12 21 7 28 7 27 7 27 7 25 7 24 7 22 22 7 37 7 37 7 36 7 35 7 34 7 33 23 7 47 7 47 7 46 7 46 7 45 7 45 24 7 57 7 57 7 57 7 57 7 57 7 57 25 7 67 7 68 7 68 7 69 7 70 7 70 26 7 78 7 79 7 81 7 82 7 83 7 84 27 7 90 7 92 7 94 7 96 7 97 7 99 28 8 03 8 05 8 08 8 10 8 12 8 14 29 8 16 8 20 8 22 8 25 8 28 8 29 30 8 31 8 34 8 38 8 41 8 43 8 46 31 8 46 8 50 8 54 8 57 8 61 8 63 32 8 63 8 67 • 8 70 8 74 8 78 8 81 33 8 79 8 84 8 88 8 92 8 .96 8 99 34 8 97 9 02 9 07 9 10 9 15 9 18 35 9 15 9 21 9 25 9 30 9 34 9 37 36 9 34 9 40 9 45 9 60 9 54 9 58 37 9 54 9 60 9 65 9 70 9 75 9 81 38 9 75 9 HI 9 86 9 92 9 99 10 11 39 9 97 10 03 10 09 10 16 10 29 10 48 40 10 19 10 26 10 34 10 48 10 67 10 91 41 10 43 10 52 10 66 ' 10 87 11 12 11 44 42 10 71 10 85 11 07 11 33 11 66 12 01 43 11 05 11 27 11 54 11 88 12.24 12 63 44 11 48 11 76 12 11 12 48 12 88 13 30 ■ 45 11 98 12 35 12 73 13 14 13 57 14 03 46 12 59 12 98 13 41 13 85 14 32 14 82 47 13 24 13 68 14 14 14 62 15 14 15 68 48 13 96 14 43 14 93 15 46 16 03 16 62 49 14 74 15 '25 15 80 16 38 16 99 17 61 50 15 59 16 15 16 75 17 38 18 02 18 71 51 i 16 51 17 13 17 78 18 43 19 14 19 88 52 17 53 18 19 18 87 19 60 20 36 21 12 . 53 18 63 19 33 20 08 20 85 21 64 22 46 54 19 81 20 58 21 38 22 18 23 03 23 92 55 21 11 21 93 22 75 23 63 24 54 25 55 ■^ COST OF INSURANCE. 49 TABLE XXII. Compound Interest. Amount ot One Dollar improved at Compound Interest for any number of years not exceeding buy. Knd of Year 4 Per Cent. a Per Cent. 6 Per Cent. 7 Per Cent. 8 Per Cent. 1 $1.0400 $1.0500 $1.0600 $1.0700 $1.0800 2 1.0816 1.1026 1.1336 1.1449 ♦ 1.1664 3 1.1349 1.1576 1.1910 1.2250 1.3597 4 1.1699 1.2155 12635 1.3108 1.3605 5 1.2167 1.3763 1.3882 1.4026 1.4693 6 1.2653 1.3401 1.4185 1.5007 1.5869 7 1.3159 1.4071 1.5036 1.6058 1.7138 8 1.3686 1.4775 1.5938 1.7382 1.8509 9 1.4233 1.5513 1.6895 1.8385 19990 10 1.4802 1.6289 1.7908 1.9672 3.1589 11 15395 1.7103 1.8983 2.1049 2 3316 13 1.6010 1.7959 2.0122 2.2522 2.5182 13 1.6651 18856 2.1S29 2.4098 2.7196 14 1.7317 1.9799 2.3609 3.5785 2.9372 15 1.8009 2.0789 3.3966 2.7590 3.1722 16 1.8730 2.1829 2.5404 3.9522 3.4259 17 1.9479 2.3920 2.6^928 3.1588 3.7000 18 2.0258 2.4066 2.8543 3.3799 3 9960 19 2.1068 2.5370 3.0256 3 6165 4.3157 20 2.1911 3.6533 3.2071 3.8697 4 6610 21 2.3788 2.7860 3.3996 4.1406 5.0338 22 2.3699 2.9353 3.6035 4.4304 5.4365 23 3.4647 3.0715 3.8197 4.7405 5.8715 34 2.5633 3 3351 4.0489 5.0734 6.3412 25 2.6658 3.3864 4.2919 5.4374 6.8485 26 3.7725 3.5557 4.5494 5.8074 7.3964 27 3.8834 3.7335 4.8328 6.2139 7.9881 28 3.9987 3.9301 5.1117 6.6488 8 6271 39 3.1187 4.1161 5.4184 7.1143 . 9.3178 30 3.3434 4.3219 5.7435 7.6123 10.0637 31 3.3731 4.5380 6.0881 8.1451 10.8677 32 3.5081 4.7649 6.4534 8.7153 11.7371 33 3.6484 5 0033 6.8406 9.3253 12.6760 34 3.7943 5.2533 7.3510 9.9781 13.6901 35 3.9461 5.5160 7.6861 10.6766 14.7853 36 4.1039 5.7918 8.1473 11.4239 15.9682 37 4.2681 6.0814 8.6361 12.2236 17.2456 38 4.4388 6.3855 9.1543 13.0793 18.6253 39 4.6164 6 7048 9 7035 13.9948 20.1153 40 4 8010 7.0400 10.3857 14.9745 31.7245 41 4.9931 7.3920 10.9029 16.0227 23.4625 42 5.1928 7.7616 11.5570 17.1448 25.3395 43 5.4005 81497 12.3505 18.3444 27.3666 44 5.6165 8.5573 13.9855 19.6285 29.5560 45 5.8413 8.9850 13.7646 21.0025 31.9204 46 6.0748 9.4843 14.5905 22.4726 34.4741 47 6 3178 9.9060 15 4659 34.0457 87.2320 48 6.5705 104013 16 3939 35.7389 40 2106 49 6.8333 10.9313 173775 27.5299 43.4274 50 7.1067 11.4674 18.4203 29.4570 46.9016 4 50 COST OF INSUBANCE. i TABLE XXIII. Compound Interest. Amount of One Dollar per annum (paid in advance , at Compound Interest, for any | | number of years not exceedi ng fifty. End of Year 4 Per Cent. 6 Per Cent. 6 Per Cent. 1 Per Cent. 8 Per Cent. 1 si.oloo $1.0500 $1.0600 $1.0700 $1.0800 2 2.1216 2.1525 2.1836 2.2149 2.2464 3 3.2465 3.3101 3.3746 3.4399 3.5061 4 4.4163 4.5256 4.6371 4.7507 4.8666 5 5.6330 5.8019 5 9753 6.1533 6.8359 , 6 6.8983 7.1420 7.3938 7.6540 7.9228 7 8 2142 8.5491 8.8975 9 2598 9.6366 8 9.5828 10.0266 10.4913 10.9780 11 4876 9 11.0061 11.5779 12.1808 12.8164 13.4866 10 12.4864 13.2068 18.9716 14.7836 15.6455 11 14.0258 14.9171 15.'^699 16.8885 17.9771 12 15.6268 16.7130 17.8821 19.1406 20.4953 13 17.2919 18.5986 20.0151 21.5505 28.2149 14 19.0236 20.5786 22.2760 24.1290 26.1521 15 20.8245 22.6575 24.6726 26.8881 29.8248 16 22.6975 24.8404 27.2199 29.8402 82.7502 17 24.6454 27.1324 29.9057 82.9990 86.4502 18 26.6712 29.5390 32.7000 36.3790 40.4463 19 28,7781 32.0660 35.7856 39.9955 40.7620 20 30.9692 34.7198 38.9927 48.8652 49.4229 21 33.2480 37.5052 42.3923 48.0057 54.4568 22 85.6179 40.4305 45.9958 52.4361 59.8933 23 38.0826 43.5020 49.8156 57.1767 65.7648 24 40.6459 46.7271 53.8645 62 2490 72.1059 25 43.3117 50.1135 58.1564 67.6765 78.9544 26 46.0824 58.6691 62.7058 73.4838 86.3508 27 48.9676 57.4026 67.5281 79.6977 94.8388 28 51.9663 61.3227 72.6398 86.3465 102.9659 29 55.0849 65.4888 78 0582 93.4608 112.2832 30 58.3283 69.7608 838017 101.0780 122.8459 31 61.7015 74.2988 89.8998 109.2182 138.2135 32 65.2095 79.0638 96.3432 117.9834 144.9506 33 68.8579 84.0670 103.1838 127.2588 157.6267 34 7^.6522 89 8203 110.4348 187.2369 171.3168 35 76.5983 94.8863 118.1209 147.9135 186.1021 36 80.7022 100.6281 126.2681 159.3874 202.0708 87 84.9703 106 7095 134.9042 171.5610 219.3159 38 89.4092 113.0950 144.0585 184.6408 237 9412 39 94.0255 119.7998 158.7620 198.6351 258.0565 40 98.8265 126.8398 164.0477 218.6096 279.7810 41 103.8196 184.2318 174.9505 229.6822 303.2435 42 109.0124 141.9933 186.5076 246.7765 328.5830 43 114.4129 150.1430 19^7580 265.1209 355.5*496 44 120.0294 158.7002 211.7435 284.7493 385.5056 45 125.8708 167.6852 225.5081 305.7518 417.4261 46 181.9454 177.1194 240.0986 328.2244 451.9002 47 138.2632 187.0254 255.5645 352.2701 489.1322 48 1448337 197.4267 271.9584 377.9990 629.8427 49 151.6671 208.3480 289.3359 405.5289 572.7702 50 158.7738 219.8154 307.7561 434.9860 619.6718 ! _u COST OF INSURANCE. 51 TABLE XXIV. Discount. Present Value of One Dollar due at the end of any number of years not exceeding fifty. End of Year 1 3 3 4 5 6 7 8 9 10 11 13 13 14 15 16 17 18 19 20 21 22 23 24 25 36 37 28 39 80 31 32 33 34 35 36 37 38 39 40 41 43 43 44 45 46 47 48 49 50 .961538 .924556 .888996 .854804 .821927 .790315 .709918 .730690 .702587 .675564 .649581 .624597 .600574 .577475 .555265 .533908 .513373 .493628 .474643 .456387. .438834 .431955 .405726 .390121 .375117 .860689 .846817 .333477 .320651 .308819 .296460 .285058 .374094 .263552 .353415 .243669 .334297 .335385 .316621 .208389 .200278 .192575 .185168 .178046 .171198 .164614 .158388 .153195 .146341 .140713 .952381 .907029 .863838 .833702 .783526 .746215 .710681 .676839 .644609 .613913 .584679 .556837 .530831 .505068 .481017 .458113 .486397 .415531 .395734 .376889 .358943 .341850 .335571 .810068 .295308 .381241 .367848 .355094 .343946 .331377 .320359 .309866 .199873 .190355 .181290 .172657 .164436 .156605 .149148 .142046 .135283 .128840 .122704 .116861 .111397 .105997 .100949 .096142 .091564 .087304 6 Per Cent. .943396 .889996 .889619 .793094 .747258 .704931 .665057 .627412 .591898 .558395 .536788 .496969 .468839 .442301 .417265 .393646 .371364 .350344 .330513 .311805 .294155 .377505 .361797 .346979 .333999 .319810 .307368 .195630 .184557 .174110 .164355 .154957 .146186 .137913 .130105 .132741 .115793 .109239 .103056 .097222 .091719 .086527 .081680 .077009 .072650 ,068.538 .064658 .060998 .057546 .054288 7 Per Cent. 8 Per Cent. .934579 .873489 .816398 .763895 .713986 ■666342 .622750 .582009 .543934 .508349 .475093 .444012 .414964 .887817 .362446 .338735 .316574 .295864 .376508 .358419 .341513 .325713 .310947 .197147 .184349 .172195 .160980 .150403 .140563 .131867 .123773 .114741 .107335 .100319 .098668 .087535 .081809 .076457 .071455 .066780 .062412 .058339 .054513 .050946 .047613 .044499 .041587 .088867 .036324 .033948 .925926 .857889 .798832 .735030 .680583 .630170 .583490 .540369 .500349 .463193 .438888 .397114 .367698 .340461 .315343 .391890 .270269 .250249 .331713 .314548 ,198656 .183941 .170315 .157699 .146018 .135302 .135187 .115914 .107328 .099377 .093016 .085200 .078889 .073045 .067685 .062625 .057986 .053690 .049718 .046031 .042631 .089464 .036541 .088834 .031338 .039007 .036859 .034869 .033037 .031321 52 COST OF INSURANCE. TABLE XXV. The Present Value of One Dollar Per aiiinuni for any number of years to fifty, at 4, 5, 6, 7, and 8 per cent, interest. ^ . 1 Yrs. 4 Per Cent, 6 Per Cent. 6 Per Cent. 7 Per Cent. 8 Per Cent. 1 .961538 .952381 .943396 .934579 .925926 2 1.886095 1.859410 1 833393 1.808018 1.788265 3 2.775091 2.728^48 2.673012 2.624316 2.577097 4 3.629895 3.545951 3.465106 3.887211 3.313127 5 4.451822 4.329477 4.212364 4.100197 8.992710 6 5.242137 5.075692 4 917324 4.766540 4.622880 7 6.002055 5.786373 5.582381 5.389289 5.206370 8 6.732745 6.463213 6.209794 5.971299 5.746639 9 7.485333 7.107822 6.801692 6.515232 6.346888 JO 8.110896 7 721735 7.360087 7.023582 6.710081 11 8 760477 8 306414 7.886875 7.498074 7.138964 12 9.385074 8.863252 8.383844 7.942686 7.536078 13 9.985648 9.393573 8.852683 8.357651 7.908771) 14 10.563123 9.898641 9.294984 8.745468 8.244237 15 11.118387 10.379658 9.712349 9.107914 8.559479 16 11.652296 10.837770 10.105S95 9.446649 8.851369 17 12.165669 11.2740()6 10.477260 9.763323 9.121638 18 12.659297 11.689587 10.827603 10.059087 9 371887 19 13.133939 12.085321 11.158116 10.385595 9.603599 20 13.590326 12.462210 11.469921 10.594014 9.818147 21 14.029160 13.821153 11.764077 10.835537 10.016803 23 14.451115 13.163003 12.041582 11.061241 10.200744 23 14.856842 18.488574 12 303279 11.272187 10 3710.59 24 15.246963 13.798643 12.550358 11.469334 10.528758 25 15.622080 14.093945 12.783356 11.653583 10.674776 26 15.982769 14.375185 13.003166 11.835779 10.809978 27 16.329586 14.648034 13.210534 11.986709 10.935165 28 16.663063 14.898137 13.406164 12.137111 11.051078 29 16.983715 15.141074 13.590721 13.277674 11.158406 30 17.292033 15.372451 13.764831 12.409041 11.257783 31 17.588494 15.592811 13 929086 12.531814 11.349799 32 17.873552 15.802677 14.084043 12.646555 11.434999 33 18.147646 16.003549 14.230230 12.758790 11.513888 34 18.411198 16.192904 14.868141 12.854009 11.516934 35 18.664613 16.374194 14.498346 12.947672 11.654568 36 18.908282 16.546853 14 620987 13.035208 11.717193 37 19.143579 16.711287 14.7867S0 13.117017 11.775179 38 19.367864 16.867898 14.846019 13.193473 11.828869 39 19.584485 17.017041 14.949075 13.264928 11.878583 40 19.793774 17 159086 15.046297 13.831709 11.924613 41 19.993053 17.294368 15.138016 13 394120 11.967235 43 20.185627 17.428208 15.224543 13.452449 12 006699 43 30 370795 17545912 15.306173 13.506962 12.048240 44 20 548841 17.663773 15.383182 18.557908 12.077074 45 30 720'J40 17.774070 15.455832 13.605522 12.108403 46 30.884654 17.880067 15,524370 13 650020 12.137409 47 21.042936 17.981016 15.589028 13.691008 12.164267 48 21.195131 18.077158 15.650027 13 730474 12.189136 49 21.341472 18 168732 15.707572 13 766799 12.212163 50 21.482185 18.255935 15.761861 13 800746 12.233485 ( 1 "I! J